Dual condensate and QCD phase transition
Bo Zhang, Falk Bruckmann, Zoltán Fodor, Christof Gattringer, and Kálmán K. Szabó
Citation: AIP Conf. Proc. 1343, 170 (2011); doi: 10.1063/1.3574966 View online: http://dx.doi.org/10.1063/1.3574966
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Dual condensate and QCD phase transition
Bo Zhang (Speaker)
∗, Falk Bruckmann
∗, Zoltán Fodor
†, Christof Gattringer
∗∗and Kálmán K. Szabó
†∗Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany
†Department of Physics, University of Wuppertal, Gaußstr. 20, D-42119, Germany
∗∗Institut für Physik, Universität Graz, Universitätsplatz 5, A-8010 Graz, Austria
Abstract. The dual condensate is a new QCD phase transition order parameter, which connnects confinement and chiral symmetry breaking as different mass limits. We discuss the relation between the fermion spectrum at general boundary conditions and the dual condensate and show numerical results for the latter from unquenchedSU(3)lattice configurations.
INTRODUCTION
The QCD phase transition manifests itself in two phe- nomena, deconfinement and chiral symmetry restoration.
The conventional order parameter for (de)confinement is the Polyakov loop, the straight loop in the time direction.
After a suitable renormalization, it is related to the free energy of a static quark viahtrPi ∼e−βF (β =1/kBT).
The Polyakov loop is small in the confined phase (and exactly vanishes in the quenched case because of center symmetry) and increases above the critical temperature.
The chiral condensate, on the other hand, is an order parameter for chiral symmetry breaking in the massless limit, as it is not invariant under chiral transformations.
In the chirally broken phase, the chiral condensate is finite, while it decays above the restoration temperature.
Lattice simulations at physical quark masses have re- vealed the QCD phase transition to be a crossover with pseudo-critical temperatures of 157±4 MeV for the chi- ral susceptibility and 170±5 MeV for the Polyakov loop in [1] (see also [2]).
The dual condensate [3] connects Polyakov loop and chiral condensate as two different mass limits, thus it also relates confinement and chiral symmetry breaking.
It is therefore particularly interesting to see what one can learn from the dual condensate about the physi- cal crossover. We here improve previous results [4] and study the dual condensate on theNf =2+1 staggered dynamical configurations of [1].
DUAL CONDENSATE
The physical boundary condition for a fermion field at fi- nite temperature is anti-periodic:ψ(t+β,~x) =−ψ(t,~x).
With ‘quark condensate’ we refer to the expectation value Σ(m) = V1hTr[(m+D−)−1]i with this boundary condition.
We here also consider general boundary conditions [5]
ψ(t+β,~x) =eiϕψ(t,~x), (1) giving the general quark condensate
Σ(m,ϕ) = 1
Vhtr[(m+Dϕ)−1]i= 1 V
∑
λϕ
1
m±iλϕ, (2) where iλϕ are the eigenvalues of the massless Dirac operator with these boundary condition (the physical boundary conditionϕ=π is among them).
FIGURE 1. Examples of closed loops on the lattice (with time running upwards). The red links geteiϕ factors from the implementation of general boundary conditions, Eqn. (4). The green lines have winding number one.
The dual condensate is defined as the first Fourier component of the general quark condensate with respect to the boundary phaseϕ:
Σ˜1(m) = Z 2π
0
dϕ
2π e−iϕΣ(ϕ) = Z 2π
0
dϕ 2πV
∑
λϕ
e−iϕ m±iλϕ
. (3)
The interpretation of this quantity is simplest in a lat- tice context. One can implement the boundary conditions (1) by multiplying a factoreiϕ to temporal links in one, say the last, time slice
U0(t=Nta) =⇒eiϕU0(t=Nta). (4)
The IX International Conference on Quark Confinement and the Hadron Spectrum - QCHS IX AIP Conf. Proc. 1343, 170-172 (2011); doi: 10.1063/1.3574966
© 2011 American Institute of Physics 978-0-7354-0899-9/$30.00
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10 20 30 40 50 60 70 0.05
0.1 0.15 0.2 0.25
λ[MeV]
10 20 30 40 50 60 70
0.0005 0.001 0.0015 0.002 0.0025 0.003
λ[MeV]
FIGURE 2. Accumulated contributions of eigenvalues to the quark condensateΣ(m,π)(top) and the dual condensate ˜Σ(m) (bottom), both in GeV3, atT=172 MeV andm=100 MeV.
The general quark condensateΣ(m,ϕ)is gauge invari- ant and as such is composed of the contributions from all kinds of closed loops. These loops receive different powers ofeiϕ factors, see Fig. 1. Now the dual conden- sate as the first Fourier component ofΣ(m,ϕ)(see (3)), picks out the contributions from all the loops with one eiϕfactor. These are loops winding once in the temperal direction, hence the dual condensate can be viewed as a
‘dressed Polyakov loop’.
In a similar way dual observables can be constructed for arbitrary gauge invariant objects (cf. [6]).
The conventional infinitely thin Polyakov loop is in- cluded in the set of loops that wind once and dominates in the limit of large probe massm(which can be seen through an expansion in 1/m). In this limit, however, more UV eigenvalues contribute to the sum in (3) [3].
We here consider all quantities unrenormalized (in [7]
we demonstrated that the (quenched) dressed Polyakov loop has only a mild dependence on the lattice spacing).
NUMERICAL RESULTS
We use dynamical improved staggered fermion config- urations from [1] on lattices of size 8×243, for tem- peratures ranging from 78 MeV to 890 MeV and lat- tice spacings from 0.282 fm to 0.028 fm. We compute 500 to 1000 lowest eigenvalues ofDfor 16 or 8 different boundary conditionsϕ∈[0,2π]with ARPACK. We cur- rently have completed the spectrum calculations for 20 to 35 configurations at temperatures between 100 MeV and 200 MeV.
2 4 6 8 10
42 44 46 48 50 52 54
λ[MeV]
250 500 750 1000 1250 1500 50
100 150 200 250 300
λ[MeV]
FIGURE 3. Distribution of the lowest eigenvalues for the confined phase (T =78 MeV, top) and the deconfined phase (T =892 MeV, bottom). We compare histograms forϕ=0 (red dashed) andϕ=π(blue).
The first problem we investigate is the convergence of the sums (2) and (3) when truncated to the number of available eigenvalues (in physical units). As the contri- bution of a±iλ pair to the condensates is 2m/λ2+m2, it is clear that this contribution decays forλmand only the lowest part of the spectrum contributes.
For the dual condensate there is an additional effect because it only probes the difference in the response of the spectra to changing boundary conditions. A strong response is manifest only in the IR spectrum [8, 6], as can be seen in Fig. 3, and for dual condensates thus only the IR contributes. Fig. 2 illustrates this effect: when using the available spectrum, the physical chiral condensate from (2) has not converged, while the dual condensate has due to the additional Fourier transform in (3).
Fig. 4 shows the (unrenormalized) general quark con- densate as a function of the boundary angleϕat different temperatures. It is flat at low temperature and depends strongly onϕfor high temperatures. Similar results were found also in non-lattice approaches [9].
Correspondingly, the dual condensate is small at low temperature and larger at high temperatures. It should serve as an order parameter for deconfinement, as the Polyakov loop does. In the quenched case this statement can be made exact because of the same behavior under center transformations [3]. Here both quantities have at least the same qualitative behavior.
In Fig. 5 we show our results for the absolute value of the unrenormalized dual condensate as a function of tem- perature, compared to the conventional Polyakov loop.
We also plot the negative logarithms of both divided by
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150
250
350
T@MeVD
0 Π
2 Π
3Π
2 2Π
j
0 0.2 0.4 0.6 0.8
150
250
350
T@MeV
0 0.2 0.4 0.6 0.8
0 1 2 3 4 5 6
0.0025 0.005 0.0075 0.01 0.0125 0.015
0 1 2 3 4 5 6
1 2 3 4
ϕ ϕ
FIGURE 4. The general quark condensateΣ(m,ϕ)inGeV3 as a function of temperature and boundary angle with m= 1 MeV (top). We remark that for largeT andϕ∼πwe expect further correlations until full convergence. The lower panels zoom into the confined phase (left,T=78 MeV,m=100 MeV) and the deconfined phase (right,T =740 MeV,m=10 MeV), respectively (each for a single configuration).
the inverse temperature (Fig. 6). For the Polyakov loop the latter has the interpretation of the free energy of an infinitely heavy quark. In analogy to that we might view the same quantity from the dressed Polyakov loops with mass parametermas the free energy of a test quark with finite massm.
All of these quantities show an order parameter be- havior in the temperature range of 100 to 200 MeV. In the future we want to identify the critical temperatures (through inflection points and susceptibilities) and study their mass dependence.
We thank Szabolcs Borsanyi for useful correspon- dence. F.B. and B.Z. are supported by DFG (BR 2872/4- 2).
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75 100 125 150 175 200 225 250 0
0.01 0.02 0.03 0.04 0.05
T[MeV]
75 100 125 150 175 200 225 250 0.01
0.02 0.03 0.04 0.05 0.06 0.07
T[MeV]
FIGURE 5. The dual condensate ˜Σ(m) in GeV3 at m= 60 MeV (top) and the Polyakov loop (bottom) as a function of temperature.
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0.35 0.4 0.45 0.5 0.55
T[MeV]
75 100 125 150 175 200 225 250 0.2
0.3 0.4 0.5 0.6
T[MeV]
FIGURE 6. The ‘free energy’−log ˜Σ/β from the dual con- densate at m=60 MeV (top) and from the Polyakov loop (−log|htrPi|/β, bottom) vs. temperature.
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